Let's break down the situation to find out when the pool would be completely filled with water.

Given:

• Pipe A can fill the pool in 4 hours.
• Pipe B can fill the pool in 9 hours.
• Pipe C can empty the pool in 6 hours.

You opened all three pipes at 3:00 PM and closed them at 3:30 PM. Afterward, only the filling pipes (A and B) remain open.

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Step 1: Work done by each pipe in 1 hour

• Pipe A's rate of filling = $\frac{1}{4}$ of the pool per hour.
• Pipe B's rate of filling = $\frac{1}{9}$ of the pool per hour.
• Pipe C's rate of emptying = $\frac{1}{6}$ of the pool per hour.

Step 2: Net rate of filling with all three pipes open

With all three pipes open, the net rate of filling is: $\text{Net rate} = \left( \frac{1}{4} + \frac{1}{9} - \frac{1}{6} \right)$ We need to simplify this:

$\frac{1}{4} = 0.25, \quad \frac{1}{9} \approx 0.111, \quad \frac{1}{6} \approx 0.167$ $\text{Net rate} = 0.25 + 0.111 - 0.167 = 0.194 \text{ (approximately)}.$

So, the net rate of filling when all three pipes are open is approximately 0.194 of the pool per hour.

Step 3: Work done in the first 30 minutes

In the first 30 minutes (or 0.5 hours), the amount of the pool filled with all three pipes open is: $\text{Amount filled} = 0.194 \times 0.5 = 0.097 \text{ (approximately)}.$ So, about 9.7% of the pool is filled in the first 30 minutes.

Step 4: Work done by pipes A and B after 3:30 PM

After 3:30 PM, only pipes A and B are open, and their combined rate is: $\text{Rate of A and B} = \frac{1}{4} + \frac{1}{9} = \frac{9 + 4}{36} = \frac{13}{36} \text{ of the pool per hour}.$

Step 5: Remaining pool to fill

The remaining part of the pool to be filled is: 1 - 0.097 = 0.903 \text{ (or 90.3% of the pool)}.

The time required to fill the remaining part is: $\text{Time} = \frac{0.903}{\frac{13}{36}} = 0.903 \times \frac{36}{13} \approx 2.5 \text{ hours}.$

Step 6: Time of completion

If you started the process at 3:00 PM, closed Pipe C at 3:30 PM, and then it took approximately 2.5 hours to fill the remaining pool, the pool would be fully filled at: $3:30 \, \text{PM} + 2.5 \, \text{hours} = 6:00 \, \text{PM}.$

Conclusion:

The pool would be completely filled at approximately 6:00 PM.

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Would you like further clarification, or have any specific questions? Here are a few related questions:

1. How do you calculate rates when multiple pipes are working together?
2. What if the pool had a different shape—would the filling time change?
3. How would a delay in closing Pipe C affect the outcome?
4. How can you calculate the total volume of water if the pool's size is known?
5. What if Pipe C had a faster rate of emptying?

Tip: Always check the rates of flow (filling or emptying) carefully when working with systems involving multiple inlets and outlets.